Visualizations of synthetic data rewards. The plots below visualize the synthetic data points \(G_1,\ldots ,G_5\)  (i.e., in 2-D embedding using UMAP [1]}for CC, MNIST, and CIFAR-10) as rewards to the corresponding parties \(1,\ldots ,5\)  over varying inverse temperature hyperparameters \(\beta =1,2,4,8\)  for different datasets and splits.
In these plots, the grey dots denote the entire synthetic dataset \(G\) , the blue dots denote party \(i\) 's original dataset \(D_i\) , and the red dots denote the synthetic data points \(G_i\)  as reward to party \(i\) . The more opaque the red dots, the earlier in time the weighted sampling algorithm samples these synthetic data points.
These plots complement our observations reported in the main paper: As the inverse temperature hyperparameter \(\beta \)  increases, the algorithm samples fewer synthetic data points \(G_i\)  but
they are more dissimilar from a party's original dataset \(D_i\) . This is consistent with our reasoning in Sec. REF  that \(\beta \)  controls the trade-off between \(|G_i|\)  vs. negative unbiased MMD. In addition, the plots show that more synthetic data points \(G_i\)  tend to be distributed to a party with a higher \(\alpha _i\)  as reward, which was empirically verified by the generally positive correlations of \(|G_i|\)  with \(\alpha _i\)  in Fig. REF .
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